Listing All Maximal Cliques in Sparse Graphs in
Near-optimal Time
David Eppstein, Maarten Löffler, and Darren Strash
arXiv:1006.5440v1 [cs.DS] 28 Jun 2010
Department of Computer Science, University of California, Irvine, USA
Abstract. The degeneracy of an n-vertex graph G is the smallest number d such
that every subgraph of G contains a vertex of degree at most d. We show that there
exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all
maximal cliques, parametrized by degeneracy. To achieve this result, we modify
the classic Bron–Kerbosch algorithm and show that it runs in time O(dn3d/3 ). We
also provide matching upper and lower bounds showing that the largest possible
number of maximal cliques in an n-vertex graph with degeneracy d (when d is a
multiple of 3 and n ≥ d + 3) is (n − d)3d/3 . Therefore, our algorithm matches the
Θ (d(n − d)3d/3 ) worst-case output size of the problem whenever n − d = Ω (n).
1
Introduction
Cliques, complete subgraphs of a graph, are of great importance in many applications. In
social networks cliques may represent closely connected clusters of actors [6,14,28,40]
and may be used as features in exponential random graph models for statistical analysis
of social networks [17,19,20,44,49]. In bioinformatics, clique finding procedures have
been used to detect structural motifs from protein similarities [26,35,36], to predict
unknown protein structures [45], and to determine the docking regions where two
biomolecules may connect to each other [22]. Clique finding problems also arise in
document clustering [3], in the recovery of depth from stereoscopic image data [29], in
computational topology [52], and in e-commerce, in the discovery of patterns of items
that are frequently purchased together [51].
Often, it is important to find not just one large clique, but all maximal cliques. Many
algorithms are now known for this problem [1,7,9,10,11,23,28,32,41,43,47] and for the
complementary problem of finding maximal independent sets [16,31,37,39,48]. One of
the most successful in practice is the Bron–Kerbosch algorithm, a simple backtracking
procedure that recursively solves subproblems specified by three sets of vertices: the
vertices that are required to be included in a partial clique, the vertices that are to be
excluded from the clique, and some remaining vertices whose status still needs to be
determined [7,9,32,35,47].
All maximal cliques can be listed in polynomial time per clique [37,48] or in a
total time proportional to the maximum possible number of cliques in an n-vertex
graph, without additional polynomial factors [15,47]. In particular, a variant of the
Bron–Kerbosch algorithm is known to be optimal in this sense [9,47]. Unfortunately this
maximum possible number of cliques is exponential [42], so that all general-purpose
algorithms for listing maximal cliques necessarily take exponential time.
We are faced with a dichotomy between theory, which states categorically that clique
finding takes exponential time, and practice, according to which clique finding is useful
and can be efficient in its areas of application. One standard way of resolving dilemmas
such as this one is to apply parametrized complexity [13]: one seeks a parameter of
instance complexity such that instances with small parameter values can be solved
quickly. A parametrized problem is said to be fixed-parameter tractable if instances with
size n and parameter value p can be solved in a time bound of the form f (p)nO(1) , where
f may grow exponentially or worse with p but is independent of n. With this style of
analysis, instances with a small parameter value are used to model problems that can
be solved quickly, while instances with a large parameter value represent a theoretical
worst case that, one hopes, does not arise in practice.
The size of the largest clique does not work well as a parameter: the maximum clique
problem, parametrized by clique size, is hard for W[1], implying that it is unlikely to
have a fixed-parameter tractable algorithm [12], and Turán graphs K nk , nk , nk ,... have (n/k)k
maximal cliques of size k forcing any algorithm that lists them all to take time larger than
any fixed-parameter-tractable bound. However, clique size is not the only parameter one
can choose. In this paper, we study maximal clique finding parametrized by degeneracy,
a frequently-used measure of the sparseness of a graph that is closely related to other
common sparsity measures such as arboricity and thickness, and that has previously been
used for other fixed-parameter problems [2,8,25,34]. We are motivated by the fact that
sparse graphs often appear in practice. For instance, the World Wide Web graph, citation
networks, and collaboration graphs have low arboricity [24], and therefore have low
degeneracy. Empirical evidence also suggests that the h-index, a measure of sparsity that
upper bounds degeneracy, is low for social networks [17]. As we show in Appendix A,
protein–protein interaction networks have low degeneracy as well. Furthermore, planar
graphs have degeneracy at most five [38], and the Barabási–Albert model of preferential
attachment [4], frequently used as a model for large scale-free social networks, produces
graphs with bounded degeneracy. We show that:
– A variant of the Bron–Kerbosch algorithm, when applied to n-vertex graphs with
degeneracy d, lists all maximal cliques in time O(dn3d/3 ).
– Every n-vertex graph with degeneracy d (where d is a multiple of three and n ≥ d +3)
has at most (n − d)3d/3 maximal cliques, and there exists an n-vertex graph with
degeneracy d that has exactly (n − d)3d/3 maximal cliques. Therefore, our variant
of the Bron–Kerbosch algorithm is optimal in the sense that its time is within a
constant of the parametrized worst-case output size.
Our algorithms are fixed-parameter tractable, with a running time of the form O( f (d)n)
where f (d) = d3d/3 . Algorithms for listing all maximal cliques in graphs of constant degeneracy in time O(n) were already known [10,11], but these algorithms had not been analyzed for their dependence on the degeneracy of the graph. We compare the parametrized
running time bounds of the known alternative algorithms to the running time of our
variant of the Bron–Kerbosch algorithm, and we show that the Bron–Kerbosch algorithm
has a much smaller dependence on the parameter d. Thus we give theoretical evidence
for the good performance for this algorithm that had previously been demonstrated
empirically.
A
B
C
D
E
F
F
C
H
I
D
E
A
G
B
G
H
I
(a)
(b)
Fig. 1. (a) A graph with degeneracy 3. (b) A vertex ordering showing that the degeneracy
is not larger than 3.
2
Preliminaries
We work with an undirected graph G = (V, E), which we assume is stored in an adjacency list data structure. We let n and m be the number of vertices and edges of G,
respectively. For a vertex v, we define Γ (v) to be the set {w | (v, w) ∈ E}, which we call
the
neighborhood of v, and similarly for a subset W ⊂ V we define Γ (W ) to be the set
T
w∈W Γ (w), which is the common neighborhood of all vertices in W .
2.1
Degeneracy
Our algorithm is parametrized on the degeneracy of a graph, a measure of its sparsity.
Definition 1 (degeneracy). The degeneracy of a graph G is the smallest value d such
that every nonempty subgraph of G contains a vertex of degree at most d [38].
Figure 1(a) shows an example of a graph of degeneracy 3. Degeneracy is also known
as the k-core number [5], width [21], and linkage [33] of a graph and is one less than
the coloring number [18]. In a graph of degeneracy d, the maximum clique size can be
at most d + 1, for any larger clique would form a subgraph in which all vertices have
degree higher than d.
If a graph has degeneracy d, then it has a degeneracy ordering, an ordering such
that each vertex has d or fewer neighbors that come later in the ordering. Figure 1(b)
shows a possible degeneracy ordering for the example. Such an ordering may be formed
from G by repeatedly removing a vertex of degree d or less: by the assumption that
G is d-degenerate, at least one such vertex exists at each step. Conversely, if G has an
ordering with this property, then it is d-degenerate, because for any subgraph H of G, the
vertex of H that comes first in the ordering has d or fewer neighbors in H. Thus, as Lick
and White [38] showed, the degeneracy may equivalently be defined as the minimum
d for which a degeneracy ordering exists. A third, equivalent definition is that d is the
minimum value for which there exists an orientation of G as a directed acyclic graph
in which all vertices have out-degree at most d [11]: such an orientation may be found
by orienting each edge from its earlier endpoint to its later endpoint in a degeneracy
ordering, and conversely if such an orientation is given then a degeneracy ordering may
be found as a topological ordering of the oriented graph.
Degeneracy is a robust measure of sparsity: it is within a constant factor of other
popular measures of sparsity including arboricity and thickness. In addition, degeneracy,
along with a degeneracy ordering, can be computed by a simple greedy strategy of
repeatedly removing a vertex with smallest degree (and its incident edges) from the
graph until it is empty. The degeneracy is the maximum of the degrees of the vertices at
the time they are removed from the graph, and the degeneracy ordering is the order in
which vertices are removed from the graph [30]. The easy computation of degeneracy
has made it a useful tool in algorithm design and analysis [11,16].
We can implement this algorithm in O(n + m) time by maintaining an array D, where
D[i] stores a list of vertices of degree i in the graph [5]. To remove a vertex of minimum
degree from the graph, we scan from the beginning of the array until we reach the first
nonempty list, remove a vertex from this list, and then update its neighbors’ degrees and
move them to the correct lists. Each vertex removal step takes time proportional to the
degree of the removed vertex, and therefore the algorithm takes linear time.
By counting the maximum possible number of edges from each vertex to later
neighbors, we get the following bound on the number of edges of a d-degenerate graph:
Lemma 1 (Proposition 3 of [38]). A graph G = (V, E) with degeneracy d has at most
d(n − d+1
2 ) edges.
2.2
The Bron–Kerbosch Algorithm
The Bron–Kerbosch algorithm [7] is a widely used algorithm for finding all maximal
cliques in a graph. It is a recursive backtracking algorithm which is easy to understand,
easy to code, and has been shown to work well in practice.
A recursive call to the Bron–Kerbosch algorithm provides three disjoint sets of
vertices R, P, and X as arguments, where R is a (possibly non-maximal) clique and
P ∪ X = Γ (R) are the vertices that are adjacent to every vertex in R. The vertices in P
will be considered to be added to clique R, while those in X must be excluded from the
clique; thus, within the recursive call, the algorithm lists all cliques in P ∪ R that are
maximal within the subgraph induced by P ∪ R ∪ X. The algorithm chooses a candidate
v in P to add to the clique R, and makes a recursive call in which v has been moved from
R to P; in this recursive call, it restricts X to the neighbors of v, since non-neighbors
cannot affect the maximality of the resulting cliques. When the recursive call returns,
v is moved to X to eliminate redundant work by further calls to the algorithm. When
the recursion reaches a level at which P and X are empty, R is a maximal clique and is
reported (see Fig. 2). To list all maximal cliques in the graph, this recursive algorithm is
called with P equal to the set of all vertices in the graph and with R and X empty.
Bron and Kerbosch also describe a heuristic called pivoting, which limits the number
of recursive calls made by their algorithm. The key observation is that for any vertex
u in P ∪ X, called a pivot, any maximal clique must contain one of u’s non-neighbors
(counting u itself as a non-neighbor). Therefore, we delay the vertices in P ∩ Γ (u) from
being added to the clique until future recursive calls, with the benefit that we make fewer
recursive calls. Tomita et al. [47] show that choosing the pivot u from P ∪ X in order
to maximize |P ∩ Γ (u)| guarantees that the Bron–Kerbosch algorithm has worst-case
running time O(3n/3 ), excluding time to write the output, which is worst-case optimal.
proc BronKerbosch(P, R, X)
1: if P ∪ X = 0/ then
2:
report R as a maximal clique
3: end if
4: for each vertex v ∈ P do
5:
BronKerbosch(P ∩ Γ (v), R ∪ {v}, X ∩ Γ (v))
6:
P ← P \ {v}
7:
X ← X ∪ {v}
8: end for
proc BronKerboschPivot(P, R, X)
1: if P ∪ X = 0/ then
2:
report R as a maximal clique
3: end if
4: choose a pivot u ∈ P ∪ X {Tomita et al. choose u to maximize |P ∩ Γ (u)|}
5: for each vertex v ∈ P \ Γ (u) do
6:
BronKerboschPivot(P ∩ Γ (v), R ∪ {v}, X ∩ Γ (v))
7:
P ← P \ {v}
8:
X ← X ∪ {v}
9: end for
Fig. 2. The Bron–Kerbosch algorithm without and with pivoting
3
The Algorithm
In this section, we show that apart from the pivoting strategy, the order in which the
vertices of G are processed by the Bron–Kerbosch algorithm is also important. By
choosing an ordering carefully, we develop a variant of the Bron–Kerbosch algorithm
that correctly lists all maximal cliques in time O(dn3d/3 ). Essentially, our algorithm
performs the outer level of recursion of the Bron–Kerbosch algorithm without pivoting,
using a degeneracy ordering to order the sequence of recursive calls made at this level,
and then switches at inner levels of recursion to the pivoting rule of Tomita et al. [47].
In the original Bron–Kerbosch algorithm, in each recursive call the vertices in P
are considered for expansion one by one (see line 4 of BronKerbosch in Figure 2). The
order in which the vertices are treated is not specified. We first analyze what happens if
we fix an order v1 , v2 , . . . , vn on the vertices of V , and use the same order consistently
to loop through the vertices of P in each recursive call of the non-pivoting version of
BronKerbosch.
Observation 1 When processing a clique R in the ordered variant of Bron–Kerbosch,
the common neighbors of R can be partitioned into the set P of vertices that come after
the last vertex of R, and the set X of remaining neighbors, as shown in Figure 3.
Our algorithm computes a degeneracy ordering of the given graph, and performs
the outermost recursive calls in the ordered variant of the Bron–Kerbosch algorithm
(without pivoting) for this ordering. The sets P passed to each of these recursive calls
will have at most d elements in them, leading to few recursive calls within each of these
X
X
R
X
R
P
Fig. 3. Partitioning the common neighbors of a clique R into the set P of later vertices and
the set X of remaining neighbors.
outer calls. Below the top level of the recursion we switch from the ordered non-pivoting
version of the Bron–Kerbosch algorithm to the pivoting algorithm (with the same choice
of pivots as Tomita et al. [47]) to further control the number of recursive calls.
proc BronKerboschDegeneracy(V , E)
1: for each vertex vi in a degeneracy ordering v0 , v1 , v2 , . . . of (V, E) do
2:
P ← Γ (vi ) ∩ {vi+1 , . . . , vn−1 }
3:
X ← Γ (vi ) ∩ {v0 , . . . , vi−1 }
4:
BronKerboschPivot(P, {vi }, X)
5: end for
Fig. 4. Our algorithm.
Lemma 2. The Bron–Kerbosch algorithm using the Tomita et al. pivoting strategy
generates all and only maximal cliques containing all vertices in R, some vertices in P,
and no vertices in X, without duplication.
Proof. See Tomita et al. [47].
t
u
Theorem 1. Algorithm BronKerboschDegeneracy generates all and only maximal
cliques without duplication.
Proof. Let C be a maximal clique, and v its earliest vertex in the degeneracy order. By
Lemma 2, C will be reported (once) when processing v. When processing any other
vertex of C, v will be in X, so C will not be reported.
t
u
To make pivot selection fast we pass as an additional argument to BronKerboschPivot
a subgraph HP,X of G that has P ∪ X as its vertices; an edge (u, v) of G is kept as an edge
in HP,X whenever at least one of u or v belongs to P and both of them belong to P ∪ X.
The pivot chosen according to the pivot rule of Tomita et al. [47] is then just the vertex
in this graph with the most neighbors in P.
Lemma 3. Whenever BronKerboschDegeneracy calls BronKerboschPivot it can form
HP,X in time O(d(|P| + |X|)).
Proof. The vertex set of HP,X is known from P and X. Each edge is among the d outgoing
edges from each of its vertices. Therefore, we can achieve the stated time bound by
looping over each of the d outgoing edges from each vertex in P ∪ X and testing whether
each edge meets the criterion for inclusion in HP,X .
t
u
The factor of d in the time bound of Lemma 3 makes it too slow for the recursive
part of our algorithm. Instead we show that the subgraph to be passed to each recursive
call can be computed quickly from the subgraph given to its parent in the recursion tree.
Lemma 4. In a recursive call to BronKerboschPivot that is passed the graph HP,X as an
auxiliary argument, the sequence of graphs HP∩Γ (v),X∩Γ (v) to be passed to lower-level
recursive calls can be computed in total time O(|P|2 (|P| + |X|)).
Proof. It takes O(|P| + |X|) time to identify the subsets P ∩ Γ (v) and X ∩ Γ (v) by
examining the neighbors of v in HP,X . Once these sets are identified, HP∩Γ (v),X∩Γ (v) may
be constructed as a subgraph of HP,X in time O(|P|(|P| + |X|)) by testing for each edge
of HP,X whether its endpoints belong to these sets. There are O(|P|) graphs to construct,
one for each recursive call, hence the total time bound.
t
u
Lemma 5 (Theorem 3 of [47]). Let T be a function which satisfies the following recurrence relation:
(
maxk {kT (p − k)} + d p2 if p > 0
T (p) ≤
e
if p = 0
Where p and k are integers, such that p ≥ k, and d, e are constants greater than zero.
Then, T (p) ≤ maxk {kT (p − k)} + d p2 = O(3 p/3 ).
Lemma 6. Let v be a vertex, Pv , be v’s later neighbors, and Xv be v’s earlier neighbors.
Then BronKerboschPivot(Pv , {v}, Xv ) executes in time O((d + |Xv |)3|Pv |/3 ), excluding
the time to report the discovered maximal cliques.
Proof. Define D(p, x) to be the running time of BronKerboschPivot(Pv , {v}, Xv ), where
p = |Pv |, and x = |Xv |. We show that D(p, x) = O((d + x)3 p/3 ). By the description of
BronKerboschPivot, D satisfies the following recurrence relation:
(
maxk {kD(p − k, x)} + c1 p2 (p + x) if p > 0
D(p, x) ≤
c2
if p = 0
where c1 and c2 are constants greater than 0.
Since our graph has degeneracy d, the inequality p + x ≤ d + x always holds. Thus,
D(p, x) ≤ max{kD(p − k, x)} + c1 p2 (p + x)
k
kD(p − k, x)
2
≤ (d + x) max
+ c1 p
d +x
k
≤ (d + x) max{kT (p − k)} + c1 p2
k
= O((d + x)3 p/3 ) by letting d = c1 , e = c2 in Lemma 5
t
u
Theorem 2. Given a n-vertex graph G with degeneracy d, our algorithm reports all
maximal cliques of G in time O(dn3d/3 ).
Proof. For each initial call to BronKerboschPivot for each vertex v, we first spend time
O(d(|Pv | + |Xv |)) to set up subgraph HPv ,Xv . Over the entire algorithm we spend time
∑ O(d(|Pv | + |Xv |)) = O(dm) = O(d 2 n)
v
setting up these subgraphs. The time spent performing the recursive calls is
∑ O((d + |Xv |)3|Pv |/3 ) = O((dn + m)3d/3 ) = O(dn3d/3 ),
v
and the time to report all cliques is O(dµ), where µ is the number of maximal cliques.
We show in the next section that µ = (n − d)3d/3 in the worst case, and therefore we
take time O(d(n − d)3d/3 ) reporting cliques in the worst case. Therefore, the algorithm
executes in time O(dn3d/3 ).
t
u
This running time is nearly worst-case optimal, since there may be Θ ((n − d)3d/3 )
maximal cliques in the worst case.
4
Worst-case Bounds on the Number of Maximal Cliques
Theorem 3. Let d be a multiple of 3 and n ≥ d + 3. Then the largest possible number
of maximal cliques in an n-vertex graph with degeneracy d is (n − d)3d/3 .
Proof. We first show that there cannot be more than (n − d)3d/3 maximal cliques. We
then show that there exists a graph that has (n − d)3d/3 maximal cliques.
An Upper Bound. Consider a degeneracy ordering of the vertices, in which each vertex
has at most d neighbors that come later in the ordering. For each vertex v that is placed
among the first n − d − 3 vertices of the degeneracy ordering, we count the number of
maximal cliques such that v is the clique vertex that comes first in the ordering.
Since vertex v has at most d later neighbors, the Moon–Moser bound [42] applied
to the subgraph induced by these later neighbors shows that they can form at most 3d/3
maximal cliques with each other. Since these vertices are all neighbors of v, v participates
in at most 3d/3 maximal cliques with its later neighbors. Thus, the first n − d − 3 vertices
contribute to at most (n − d − 3)3d/3 maximal cliques total.
By the Moon–Moser bound, the remaining d + 3 vertices in the ordering can form at
most 3(d+3)/3 maximal cliques. Therefore, a graph with degeneracy d can have no more
than (n − d − 3)3d/3 + 3(d+3)/3 = (n − d)3d/3 maximal cliques.
A Lower Bound. By a simple counting argument, we can see that the graph Kn−d,3,3,3,3,...
contains (n − d)3d/3 maximal cliques: Each maximal clique must contain exactly one
vertex from each disjoint independent set of vertices, and there are (n − d)3d/3 ways of
forming a maximal clique by choosing one vertex from each independent set. Figure 5
shows this construction for d = 6. We can also see that this graph is d-degenerate since
in any ordering of the vertices, the first vertex must have d or more later neighbors, and
in any ordering where the n − d disjoint vertices come first, these first n − d vertices have
exactly d later neighbors, and the last d vertices have fewer later neighbors.
t
u
Relatedly, a bound of (n − d + 1)2d on the number of cliques (without assumption
of maximality) in n-vertex d-degenerate graphs was already known [50].
Fig. 5. The lower bound construction for d = 6, consisting of a Moon–Moser graph of size
d on the right (blue vertices) and an independent set of n − d remaining vertices that are
each connected to all of the last d vertices.
5
Comparison with Other Algorithms
Chiba and Nishizeki [10] describe two algorithms for finding cliques in sparse graphs.
The first of these two algorithms reports all maximal cliques using O(am) time per
clique, where a is the arboricity of the graph, and m is the number of edges in G. The
arboricity is the minimum number of edge-disjoint spanning forests into which the graph
can be decomposed [27]. The degeneracy of a graph is closely related to arboricity:
a ≤ d ≤ 2a − 1. In terms of degeneracy, Chiba and Nishizeki’s algorithm uses O(d 2 n)
time per clique. Combining this with the bound on the number of cliques derived in
Section 4 results in a worst-case time bound of O(d 2 n(n − d)3d/3 ). For constant d, this
is a quadratic time bound, in contrast to the linear time of our algorithm.
Another algorithm of Chiba and Nishizeki [10] lists cliques of order l in time
O(lal−2 m). It can be adapted to enumerate all maximal cliques in a graph with degeneracy d by first enumerating all cliques of order d + 1, d, . . . down to 1, and removing cliques that are not maximal. Applying their algorithm directly to a d-degenerate
graph takes time O(ld l−1 n). Therefore, the running time to find all maximal cliques is
∑1≤i≤d+1 O(ind i−1 ) = O(nd d+1 ). Like our algorithm, this is linear when d is constant,
but with a much worse dependence on the parameter d.
Chrobak and Eppstein [11] list triangles and 4-cliques in graphs of bounded degeneracy by testing all sets of two or three later neighbors of each vertex according to
a degeneracy ordering. The same idea extends in an obvious way to finding maximal
cliques of size greater than four, by testing all subsets of later neighbors of each vertex.
For each vertex v, there are at most 2d subsets to test; each subset may be tested for
being a clique in time O(d 2 ), by checking whether each of its vertices has all the later
vertices in the subset among its later neighbors, giving a total time of O(nd 2 2d ) to list
all the cliques in the graph. However, although this singly-exponential time bound is
considerably faster than Chiba and Nishizeki, and is close to known bounds on the
number of (possibly non-maximal) cliques in d-degenerate graphs [50], it is slower than
our algorithm by a factor that is exponential in d. Our new algorithm uses this same idea
of searching among the later neighbors in a degeneracy order but achieves much greater
efficiency by combining it with the Bron–Kerbosch algorithm.
6
Conclusion
We have presented theoretical evidence for the fast performance of the Bron–Kerbosch
algorithm for finding cliques in graphs, as has been observed in practice. We observe
that the problem is fixed-parameter tractable in terms of the degeneracy of the graph,
a parameter that is expected to be low in many real-world applications, and that a
slight modification of the Bron–Kerbosch algorithm performs optimally in terms of the
degeneracy.
We explicitly prescribe the order in which the Bron–Kerbosch algorithm processes
the vertices of the graph, something that has not been considered before. Without this
particular order, we do not have a bound on the running time. It would be interesting to
determine whether a random order gives similar results, as this would further explain
the observed performance of implementations of Bron–Kerbosch that do not use the
degeneracy order.
Acknowledgments
This research was supported in part by the National Science Foundation under grant
0830403, and by the Office of Naval Research under MURI grant N00014-08-1-1015.
References
1. E. A. Akkoyunlu. The enumeration of maximal cliques of large graphs. SIAM J. Comput.
2(1):1–6, 1973, doi:10.1137/0202001.
2. N. Alon and S. Gutner. Linear time algorithms for finding a dominating set of fixed size in
degenerated graphs. Algorithmica 54(4):544–556, 2009, doi:10.1007/s00453-008-9204-0.
3. J. G. Augustson and J. Minker. An analysis of some graph theoretical cluster techniques. J.
ACM 17(4):571–588, 1970, doi:10.1145/321607.321608.
4. A.-L. Barabási and R. Albert. Emergence of scaling in random networks. Science
286:509–512, 1999, doi:10.1126/science.286.5439.509.
5. V. Batagelj and M. Zaveršnik. An O(m) algorithm for cores decomposition of networks,
2003, arXiv:cs/0310049.
6. N. M. Berry, T. H. Ko, T. Moy, J. Smrcka, J. Turnley, and B. Wu. Emergent clique formation
in terrorist recruitment. Proc. AAAI-04 Worksh. Agent Organizations. AAAI Press, 2004,
http://www.aaai.org/Papers/Workshops/2004/WS-04-02/WS04-02-005.pdf.
7. C. Bron and J. Kerbosch. Algorithm 457: finding all cliques of an undirected graph. Commun.
ACM 16(9):575–577, 1973, doi:10.1145/362342.362367.
8. L. Cai, S. Chan, and S. Chan. Random separation: A new method for solving fixed-cardinality
optimization problems. Proc. 2nd Int. Worksh. Parameterized and Exact Computation
(IWPEC 2006), pp. 239–250. Springer-Verlag, LNCS 4169, 2006, doi:10.1007/11847250 22.
9. F. Cazals and C. Karande. A note on the problem of reporting maximal cliques. Theor.
Comput. Sci. 407(1-3):564 – 568, 2008, doi:10.1016/j.tcs.2008.05.010.
10. N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput.
14(1):210–223, 1985, doi:10.1137/0214017.
11. M. Chrobak and D. Eppstein. Planar orientations with low out-degree and compaction of
adjacency matrices. Theor. Comput. Sci. 86(2):243 – 266, 1991,
doi:10.1016/0304-3975(91)90020-3.
12. R. G. Downey and M. R. Fellows. Fixed-parameter tractability and completeness II: On
completeness for W[1]. Theor. Comput. Sci. 141(1-2):109 – 131, 1995,
doi:10.1016/0304-3975(94)00097-3.
13. R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.
14. N. Du, B. Wu, X. Pei, B. Wang, and L. Xu. Community detection in large-scale social
networks. Proc. 9th WebKDD and 1st SNA-KDD 2007 Workshop on Web Mining and Social
Network Analysis, pp. 16–25, 2007, doi:10.1145/1348549.1348552.
15. D. Eppstein. Small maximal independent sets and faster exact graph coloring. J. Graph
Algorithms & Applications 7(2):131–140, 2003, arXiv:cs.DS/0011009.
16. D. Eppstein. All maximal independent sets and dynamic dominance for sparse graphs. ACM
Trans. Algorithms 5(4):A38, 2009, doi:10.1145/1597036.1597042, arXiv:cs.DS/0407036.
17. D. Eppstein and E. S. Spiro. The h-index of a graph and its application to dynamic subgraph
statistics. Proc. 11th Symp. Algorithms and Data Structures (WADS 2009), pp. 278–289.
Springer-Verlag, LNCS 5664, 2009, doi:10.1007/978-3-642-03367-4 25.
18. P. Erdős and A. Hajnal. On chromatic number of graphs and set-systems. Acta Mathematica
Hungarica 17(1–2):61–99, 1966, doi:10.1007/BF02020444.
19. O. Frank. Statistical analysis of change in networks. Statistica Neerlandica 45(3):283–293,
1991, doi:10.1111/j.1467-9574.1991.tb01310.x.
20. O. Frank and D. Strauss. Markov graphs. J. Am. Stat. Assoc. 81(395):832–842, 1986,
doi:10.2307/2289017.
21. E. C. Freuder. A sufficient condition for backtrack-free search. J. ACM 29(1):24–32, 1982,
doi:10.1145/322290.322292.
22. E. J. Gardiner, P. Willett, and P. J. Artymiuk. Graph-theoretic techniques for macromolecular docking. J. Chem. Inf. Comput. Sci. 40(2):273–279, 2000, doi:10.1021/ci990262o.
23. L. Gerhards and W. Lindenberg. Clique detection for nondirected graphs: Two new
algorithms. Computing 21(4):295–322, 1979, doi:10.1007/BF02248731.
24. G. Goel and J. Gustedt. Bounded arboricity to determine the local structure of sparse graphs.
WG 2006, pp. 159–167. Springer-Verlag, LNCS 4271, 2006, doi:10.1007/11917496 15.
25. P. A. Golovach and Y. Villanger. Parameterized complexity for domination problems on
degenerate graphs. Proc. 34th Int. Worksh. Graph-Theoretic Concepts in Computer Science
(WG 2008) 5344:195–205, 2008, doi:10.1007/978-3-540-92248-3 18.
26. H. M. Grindley, P. J. Artymiuk, D. W. Rice, and P. Willett. Identification of tertiary structure
resemblance in proteins using a maximal common subgraph isomorphism algorithm. J. Mol.
Biol. 229(3):707 – 721, 1993, doi:10.1006/jmbi.1993.1074.
27. F. Harary. Graph Theory. Addison-Wesley, Reading, MA, 1972.
28. F. Harary and I. C. Ross. A procedure for clique detection using the group matrix.
Sociometry 20(3):205–215, 1957, doi:10.2307/2785673.
29. R. Horaud and T. Skordas. Stereo correspondence through feature grouping and maximal
cliques. IEEE Trans. Patt. An. Mach. Int. 11(11):1168–1180, 1989, doi:10.1109/34.42855.
30. T. R. Jensen and B. Toft. Graph Coloring Problems. Wiley-Interscience, New York, 1995.
31. D. S. Johnson, M. Yannakakis, and C. H. Papadimitriou. On generating all maximal independent sets. Inf. Proc. Lett. 27(3):119 – 123, 1988, doi:10.1016/0020-0190(88)90065-8.
32. H. C. Johnston. Cliques of a graph—variations on the Bron–Kerbosch algorithm. Int. J.
Parallel Programming 5(3):209–238, 1976, doi:10.1007/BF00991836.
33. L. Kirousis and D. Thilikos. The linkage of a graph. SIAM J. Comput. 25(3):626–647, 1996,
doi:10.1137/S0097539793255709.
34. T. Kloks and L. Cai. Parameterized tractability of some (efficient) Y -domination variants for
planar graphs and t-degenerate graphs. Proc. International Computer Symposium, 2000,
http://hdl.handle.net/2377/2482.
35. I. Koch. Enumerating all connected maximal common subgraphs in two graphs. Theor.
Comput. Sci. 250(1-2):1 – 30, 2001, doi:10.1016/S0304-3975(00)00286-3.
36. I. Koch, T. Lengauer, and E. Wanke. An algorithm for finding maximal common
subtopologies in a set of protein structures. J. Comput. Biol. 3(2):289–306, 1996,
doi:10.1089/cmb.1996.3.289.
37. E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Generating all maximal independent
sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9(3):558–565, 1980,
doi:10.1137/0209042.
38. D. R. Lick and A. T. White. k-degenerate graphs. Canad. J. Math. 22:1082–1096, 1970,
http://www.smc.math.ca/cjm/v22/p1082.
39. E. Loukakis and C. Tsouros. A depth first search algorithm to generate the family of maximal
independent sets of a graph lexicographically. Computing 27(4):349–366, 1981,
doi:10.1007/BF02277184.
40. R. D. Luce and A. D. Perry. A method of matrix analysis of group structure. Psychometrika
14(2):95–116, 1949, doi:10.1007/BF02289146.
41. K. Makino and T. Uno. New algorithms for enumerating all maximal cliques. Proc. 9th
Scand. Worksh. Algorithm Theory, pp. 260–272. Springer-Verlag, LNCS 3111, 2004.
42. J. W. Moon and L. Moser. On cliques in graphs. Israel J. Math. 3(1):23–28, 1965,
doi:10.1007/BF02760024.
43. G. D. Mulligan and D. G. Corneil. Corrections to Bierstone’s algorithm for generating
cliques. J. ACM 19(2):244–247, 1972, doi:10.1145/321694.321698.
44. G. Robins and M. Morris. Advances in exponential random graph (p∗ ) models. Social
Networks 29(2):169–172, 2007, doi:10.1016/j.socnet.2006.08.004.
45. R. Samudrala and J. Moult. A graph-theoretic algorithm for comparative modeling of protein
structure. J. Mol. Biol. 279(1):287 – 302, 1998, doi:10.1006/jmbi.1998.1689.
46. C. Stark, B.-J. Breitkreutz, T. Reguly, L. Boucher, A. Breitkreutz, and M. Tyers. BioGRID: a
general repository for interaction datasets. Nucleic Acids Res. 34:D535–D539, 2006,
doi:10.1093/nar/gkj109.
47. E. Tomita, A. Tanaka, and H. Takahashi. The worst-case time complexity for generating all
maximal cliques and computational experiments. Theor. Comput. Sci. 363(1):28–42, 2006,
doi:10.1016/j.tcs.2006.06.015.
48. S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa. A new algorithm for generating all the
maximal independent sets. SIAM J. Comput. 6(3):505–517, 1977, doi:10.1137/0206036.
49. S. Wasserman and P. Pattison. Logit models and logistic regressions for social networks: I.
An introduction to Markov graphs and p∗ . Psychometrika 61(3):401–425, 1996,
doi:10.1007/BF02294547.
50. D. R. Wood. On the maximum number of cliques in a graph. Graphs and Combinatorics
23(3):337–352, 2007, doi:10.1007/s00373-007-0738-8.
51. M. J. Zaki, S. Parthasarathy, M. Ogihara, and W. Li. New algorithms for fast discovery of
association rules. Proc. 3rd Int. Conf. Knowledge Discovery and Data Mining, pp. 283–286.
AAAI Press, 1997, http://www.aaai.org/Papers/KDD/1997/KDD97-060.pdf.
52. A. Zomorodian. The tidy set: a minimal simplicial set for computing homology of clique
complexes. Proc. 26th ACM Symp. Computational Geometry, pp. 257–266, 2010,
http://www.cs.dartmouth.edu/~afra/papers/socg10/tidy-socg.pdf.
A
Appendix
A.1
The Degeneracy of Protein–Protein Interaction Networks
The Biological General Repository for Interaction Datasets (BioGRID) [46] 1 is a
curated database containing the interactions between proteins that have been published
in the literature, and those that have been discovered through high-throughput screening
methods. Using this interaction data, we can form a graph by creating a vertex for
each protein and an edge between two proteins that interact. Such a graph is called a
protein–protein interaction (PPI) network.
We computed the degeneracy of seven PPI networks in version 3.0.65 of the BioGRID
database. We chose to omit other networks in the BioGRID database because they contain
only a handful of known proteins and interactions. The results are summarized in Table 1.
Observe that, for each PPI network, the degeneracy is significantly lower than both the
number of vertices in the graph and the maximum degree of the graph. We therefore
conclude that the degeneracy of these PPI networks is low and that fixed-parameter
algorithms parametrized by degeneracy can be expected to perform well on these graphs.
Table 1. Graph statistics for seven PPI networks from version 3.0.65 of the BioGRID
database. For each network we list the number of vertices (n), the number of edges (m),
the maximum degree (∆ ), and the degeneracy (d).
Organism (PPI network)
Mus musculus
Caenorhabditis elegans
Arabisopsis thaliana
Drosophila melanogaster
Homo Sapiens
Schizosaccharomyces pombe
Saccharomyces cerevisae
1
http://thebiogrid.org/
n
1455
3518
1745
7282
9527
2031
6008
m
1636
6531
3098
24894
31182
12637
156945
∆
111
523
71
176
308
439
2557
d
6
10
12
12
12
34
64